Mathematics
Writing Equations
Writing equations involves expressing mathematical relationships using symbols and numbers. This process allows for the representation of various mathematical concepts, such as linear or quadratic relationships, in a concise and standardized form. Equations are used to solve problems, make predictions, and communicate mathematical ideas in a clear and precise manner.
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eBook - ePubCommon Core Standards for Middle School Mathematics
A Quick-Start Guide
- Amitra Schwols, Kathleen Dempsey, John Kendall(Authors)
- 2013(Publication Date)
- ASCD(Publisher)
Chapter 4Expressions and Equations
. . . . . . . . . . . . . . . . . . . .The introduction to the Common Core standards' high school Algebra domain provides some useful definitions to help differentiate the two mathematical terms expression and equation:An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function…. An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. (CCSSI, 2010c, p. 62)The middle school Expressions and Equations (EE) domain provides a critical bridge between content in the Operations and Algebraic Thinking domain in earlier grades and algebraic content students will encounter in high school. Figure 4.1 shows an overview of the Expressions and Equations domain's clusters and standards by grade level.
eBook - PDFAlgebra Teacher's Activities Kit
150 Activities that Support Algebra in the Common Core Math Standards, Grades 6-12
- Judith A. Muschla, Gary Robert Muschla, Erin Muschla-Berry(Authors)
- 2015(Publication Date)
- Jossey-Bass(Publisher)
3–2: (6.EE.2) WRITING AND READING ALGEBRAIC EXPRESSIONS For this activity, your students will be given phrases containing algebraic expressions. They are to determine if each expression is stated correctly. If it is incorrect, they are to correct the expression. Completing a statement at the end of the worksheet will enable students to check their work. Explain that algebraic expressions contain variables. A variable is a letter that represents a number. Discuss the examples on the worksheet. Emphasize that when writing an expression, order does not matter for addition and multiplication, but order does matter for subtraction and division. Grouping symbols indicate that a quantity must be treated as a unit. You might want to caution your students to be careful not to read the variable o as a zero. Review the directions on the worksheet. Students must correct the incorrect expressions and use the variables of the corrected expressions to complete the statement at the end. ANSWERS Answers to incorrect problems are provided. (2) s – 2 (5) ( t + 6 ) ÷ 12 (7) 4 2 + u (8) p ( 8 – 2 ) (11) ( e − 5 ) ÷ 6 (13) n – 15 (16) 8 d – 1 (17) 6 ( o + 3 ) (19) 3 u ÷ 3 (20) 22 s Your work with algebraic expressions is “stupendous.” 3–3: (6.EE.2) EVALUATING ALGEBRAIC EXPRESSIONS Your students will evaluate algebraic expressions in equations for this activity. By unscrambling the letters of their answers to find a math word, they can check their answers. Explain that an equation is a mathematical sentence that expresses a relationship between two quantities. Formulas are a special type of equation. Provide this example. The perimeter of a square is four times the length of a side. This can be written as an equation P = 4 s . Students can find the value of P if they know the length, s , of a side. If s = 12 . 5 inches, students can substitute 12.5 for s into the expression 4 s to find that P = 50 inches. Go over the directions on the worksheet with your students.
eBook - PDF- Daniel Ashlock(Author)
- 2022(Publication Date)
- Springer(Publisher)
1 C H A P T E R 1 Review of Algebra This book is a text on calculus, structured to prepare students for applying calculus to the physical sciences. The first chapter has no calculus in it at all; it is here because many students manage to get to the university or college level without adequate skill in algebra, trigonometry, or geometry. We assume familiarity with the concept of variables like x and y that denote numbers whose value is not known. 1.1 SOLVING EQUATIONS An equation is an expression with an equals sign in it. For example: x D 3 is a very simple equation. It tells us that the value of the variable x is the number 3. There are a number of rules we can use to manipulate equations. What these rules do is change an equation into another equation that has the same meaning but a different form. The things we can do to an equation without changing its meaning include the following. • Add or subtract the same term from both sides. If that term is one that is already present in the equation, we may call this moving the term to the other side. When this hap- pens, the term changes sign, from positive to negative or negative to positive. For example: x 4 D 5 This is the original equation x D 5 C 4 Add 4 to both sides (move 4 to the other side) x D 9 Finish the arithmetic • Multiply or divide both sides by the same expression. 2 1. REVIEW OF ALGEBRA For example: 3x 4 D 5 This is the original equation 3x D 5 C 4 Add 4 to both sides (move 4 to the other side) 3x D 9 Finish the addition 3x 3 D 9 3 Divide both sides by 3 x D 3 Finish the arithmetic • Apply the same function or operation to both sides. For example: p x 2 D 5 This is the original equation p x 2 2 D 5 2 Square both sides x 2 D 25 Do the arithmetic x D 27 Move 2 to the other side Knowledge Box 1.1 The rules for solving equations include: 1. Adding or subtracting the same thing from both sides. 2. Moving a term to the other side; its sign changes. 3. Multiplying or dividing both sides by the same thing.
eBook - PDFDiscrete Mathematics
Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
- Douglas E. Ensley, J. Winston Crawley(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
If we are patient enough, the logical structure of your mathematical writing will seem as natural as an argument with your sister. 81 82 Chapter 2 / A Primer of Mathematical Writing 2.1 Mathematical Writing Mathematicians today rely on the notion of symbolic logic to serve as the foundation of the system of reasoning that they bring to bear on the solution of problems. Of course, mathematics was quite successful for many centuries before symbolic logic was introduced, so it is not necessary to study symbolic logic in order to understand formal mathematics. A defining characteristic of mathematics is the thorough under- standing of the cause-and-effect relationship between properties of formally defined objects. The demonstration of this understanding is the notion of “proof” around which the mathematics literature is based. In this section we will try to understand how to recognize a proof when we see one, and we will start to develop the skill of writing them ourselves. Unlike conversational English, most mathematical statements are in the form of implications. That is, since mathematics studies relationships between formal objects, statements that effectively state, “Whenever an object has property P, then it must also have property Q ” are commonplace. These are often more succinctly written as “if p , then q ” but the nature of the English language allows a wide variety of equivalent forms. Example 1 Write each of the following statements in “if, then” form: 1. Whenever n is an even integer, 2n 3 + n is divisible by 3. 2. For every prime n, n 2 − n + 41 is also prime. 3. The sum of the interior angles of any triangle is 180 ◦ . SOLUTION 1. If an integer n is even, then 2n 3 + n is divisible by 3. 2. If a positive integer n is prime, then the number n 2 − n + 41 is prime. 3. If P is a triangle, then the sum of the interior angles of P is 180 ◦ .
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
EQUATION SOLVING Equation solving is often advocated as a good way of introducing students to algebra, often at a very early age, in the guise of 'find the missing number' or 'I am thinking of a number' problems. T: I am thinking of a number and I double it. Then I take 5 away and the answer is 7. What was my number? A: 6 T: Is that right? B: Yes. It's 2 times 6 and that's 12 and then take 5 to get 7. T: So, A, how did you work it out? A: Well, 12 take 5 is 7, so you halve 12 and get 6. T. Did anybody think about it differently? C You took 5 away so I added it back on to 7 to get 12. Then I halved it, like A. T: Good. Now D, you think of a number and multiply it by 5 and then add 7. D: That makes 52. T: So, what is D's number? It is a short step from this sort of dialogue to representing the equations algebraically as 2x -5 = 7 or 5x + 7 = 52 and then to consider more formal methods of solution. Representing such puzzles as equations is a valuable task because it involves translating a verbal statement into a mathematical form and the reverse process of interpreting the math-ematical statement is similarly important. Helping students to make this translation in each direction is often neglected because greater emphasis is given to the process of solution. Unfortunately, learning formal procedures at too early a stage can lead to confusion through students failing to understand how the procedure mirrors the mental method. It can also lead to students thinking that algebra is a harder way to do on paper something that they can easily work out in their head, which encourages them to think that algebra is of little use because they have other ways of solving the teacher's problems. It should be clear that formal procedures are only useful when it becomes difficult to see the way to a solution without a systematic method which may involve written steps.
- Alberto D. Yazon(Author)
- 2020(Publication Date)
- Society Publishing(Publisher)
Writing skills can only be learned by constructive practice; solving mathematics problems is a natural tool for increasing writing skills for students. Classroom Implications The motivating and encouraging students to learn new words, ask questions, and write their own thoughts easily happens if they are interested, explore, and Relationship between Mathematics and Literacy 177 participate in their mathematical study. The teachers should take advantage of the inherent curiosity and inquisitiveness of students to improve language skills when studying practical mathematics. The incorporation of literacy programs into courses of mathematics helps explain concepts and can make mathematics more practical and interesting. The teachers may use a diverse range of literature, including books on commerce, documents, and fiction. Choosing a book of fiction with a mathematical theme provides both information and fascinates student interest. The fiction works successfully with young learners through the integration of cognitive learning into creative stories. Inviting students to write mathematical papers about their problems or to express and defend their views on mathematics provides opportunities for clarifying their thoughts and improving communication skills. The documentation and analysis of circumstances involving mathematics and writing informative letters on social problems like the use of sampling by the census office are other ways of incorporating writing into mathematics. The National Mathematical Teachers Council lists the outstanding new literature and multimedia resources every year. For English language learners, the use of hands-on resources may improve instruction in mathematics. The communication with materials and phenomena makes it possible for English-speaking students to ask questions from the materials themselves and use materials in dialog with teachers and peers as visual aids.
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